REPRESENTATION THEORY IN HOMOTOPY AND THE EHP SEQUENCES FOR (p 1)-CELL COMPLEXES

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1 REPRESENTATION THEORY IN HOMOTOPY AND THE EHP SEQUENCES FOR (p 1)-CELL COMPLEXES J. WU Abstract. For spaces localized at 2, the classical EHP fibrations [1, 13] Ω 2 S 2n+1 P S n E ΩS n+1 H ΩS 2n+1 play a crucial role for the computations of the homotopy groups of the spheres [16, 25]. The EHP-fibrations for (p 1)-cell complexes for p > 2 are given in this article. These fibrations can be regarded as the odd prime analogue of the classical EHP-fibrations by considering the spheres as 1-cell complexes for p = 2. Some fundamental results on the theory of natural coalgebra decompositions of tensor algebras are established in this article. As a consequence of the study on the representation theory in homotopy, the new EHP-fibrations are obtained from the evaluations of the functor A min on (p 1)-cell complexes. An application to H-spaces is also given. Contents 1. Introduction 2 2. Representation Theory of The Tensor Algebra Functor The Functor A min on Ungraded Modules The Symmetric Group Module Lie(n) Representation Theory of Tensor Algebra Functor The Functor A min for Graded and Differential Graded Modules Proof of Theorem Determination of k B(V ) T (V ) Proof of Theorem The Geometry of Natural Coalgebra Decompositions Geometric Realizations Suspension Splitting Theorems Geometric Realization of Natural Coalgebra-Split Sub Hopf Algebras Hopf Invariants Proof of Theorem Proof of Theorem Self-maps of Āmin (Y ) Proof of Theorem References 62 Research is supported in part by the Academic Research Fund of the National University of Singapore. 1

2 2 J. WU 1. Introduction For spaces localized at 2, the classical EHP fibrations [1, 13] Ω 2 S 2n+1 P S n E ΩS n+1 H ΩS 2n+1 play a crucial role for the computations of the homotopy groups of the spheres [16, 25]. In this article, we will give the EHP-fibrations for (p 1)-cell complexes for p > 2, which can be regarded as the odd prime analogue of the classical EHPfibrations by considering the spheres as 1-cell complexes for p = 2. The new EHPfibrations are one of the consequences of our study on the representation theory in homotopy as in the title. These fibrations will be obtained from the evaluations of the functor A min on (p 1)-cell complexes (See Theorem 1.5). Of course the new EHP-fibrations help the computations on the homotopy groups of finite complexes. An application to H-spaces is given in Theorem 1.6 The algebraic functor A min was introduced in [20] arising from the question on the naturality of the classical Poincaré-Birkhoff-Witt isomorphism. For any ungraded module V, A min (V ) is defined to be the smallest functorial coalgebra retract of T (V ) containing V. Then the functor A min extends canonically to the cases when V is any graded module. (See Section 2 for details.) The functor A min admits the tensor-length decomposition with A min (V ) = n=0 A min n (V ), where A min n (V ) = A min (V ) T n (V ) is the homogenous component of A min (V ). By the Functorial Poincaré-Birkhoff-Witt Theorem [20, Theorem 6.5], there exists a functor B max from (graded) modules to Hopf algebras with the functorial coalgebra decomposition T (V ) = A min (V ) B max (V ) for any graded module V. The determination of A min (V ) for general V is equivalent to an open problem in the modular representation theory of the symmetric groups according to [20, Theorem 7.4], which seems beyond the reach of current techniques. However we are able to determine A min (V ) in the special cases when V even = 0 and dimv = p 1. Denote by L(V ) the free Lie algebra generated by V. Write L n (V ) for the n-th homogeneous component of L(V ). Observe that [L s (V ), L t (V )] is a submodule of L s+t (V ) under the Lie bracket of L(V ). Let n 2 L n (V ) = L n (V )/ [L i (V ), L n i (V )]. i=2 Define L k n(v ) recursively by L 1 n(v ) = L n (V ) and L k+1 n (V ) = L n ( L k n(v )). Theorem 1.1. Let the ground field k be of characteristic p > 2. Let V be a graded module such that V even = 0 and dimv = p 1. Then there is an isomorphism of coalgebras A min (V ) = E( L k p(v )), k=0 where E(W ) is the exterior algebra generated by W.

3 REPRESENTATION THEORY IN HOMOTOPY AND EHP SEQUENCES 3 An important observation is that L k p(v ) even = 0 and dim L k p(v ) = p 1 for each k 0 provided that V even = 0 and dimv = p 1. Replace V by L p (V ) in the above theorem. Then A min ( L p (V )) = E( L k p(v )) with a coalgebra decomposition k=1 A min (V ) = E(V ) A min ( L p (V )), which indicates the existence of the EHP fibrations by taking the geometric realization of the functor A min and the Hopf invariants on the functor A min. The geometric realization of the (algebraic) functor A min is given as follows: Theorem 1.2 (Theorem 4.3). There exist homotopy functors Āmin and B max from simply connected co-h spaces of finite type to spaces such that for any p-local simply connected co-h space Y the following hold: 1) Āmin (Y ) is a functorial retract of ΩY and so there is a functorial decomposition ΩY Āmin (Y ) B max (Y ). 2) On mod p homology the decomposition H (ΩY ) = H (Āmin (Y )) H ( B max (Y )) is with respect to the augmentation ideal filtration. 3) On mod p homology the associated bigraded E 0 H (Āmin (Y )) is given by E 0 H (Āmin (Y )) = A min (Σ 1 H (Y )). We give some historic remarks to the above theorem. It was first proved in [20] that this theorem holds if Y = ΣX for a p-torsion suspension X, where Āmin (Y ) is denoted as A min (X) in [20]. After solving some technical difficulties, it was proved in [22] that this result holds if Y = ΣX for any p-local path-connected space X. Then it was given in [18, Theorem 1.1] that this result holds for any simply connected coassociative co-h-spaces Y by using the methods arising from [24]. Finally after introducing new techniques it was given in [19, Theorem 1.2] that the assumption of coassociativity on Y can be removed. It should be pointed out that it is important to remove the assumption of coassociativity as it is difficult to examine whether a co-h-space Y is coassociative or not in general. Let X be a path-connected finite complex. Define b X = qdim H q (X; Z/p). Roughly speaking, b X is the summation of the dimensions of the cells in X. A direct consequence of Theorems 1.1 and 1.2 is the following homological information: Corollary 1.3. Let p > 2 and let Y be any p-local simply connected co-h space. Suppose that H odd (Y ) = 0 and dim H (Y ) = p 1. Then there is an isomorphism of coalgebras H (Āmin (Y )) = E(Σ 1 H (Y )) k=1 q=1 E(Σ pk 1 p 1 b Y p k H (Y )), where Σ pk 1 p 1 b Y p k H (Y ) = L k p(σ 1 H (Y )).

4 4 J. WU Recall that the classical Hopf invariants can be obtained from the suspension splitting of the loop suspensions. For the geometric functor Āmin, we have the following suspension splitting theorem. Theorem 1.4 (Theorem 4.8). Let Y be any p-local simply connected co-h space. Then there is a suspension splitting ΣĀmin (Y ) Ā min n (Y ) such that for each n 1. Σ 1 H (Āmin n (Y )) = A min n (Σ 1 H (Y )) Note that each Āmin n (Y ) is a co-h space because it is a retract of ΣĀmin (Y ). From the above theorem, one gets the Hopf invariant H n defined as the composite Ā min (Y ) ΩΣĀmin (Y ) Ω Ā min n (Y ) proj Ω Ã min n (Y ) r Ā min (Āmin n (Y )), where r : ΩZ Āmin (Z) is the functorial retraction for any simply connected co- H-space Z. In particular, there is a Hopf invariant H p : Ā min (Y ) Āmin (Āmin p (Y )) for any simply connected co-h space Y. By Lemma 5.3, Ā min p (Y ) Σ b Y p+1 Y provided that H odd (Y ) = 0 and dim H (Y ) = p 1. Theorem 1.5 (EHP Fibration). Let p > 2 and let Y be any p-local simply connected co-h space. Suppose that Hodd (Y ) = 0 and dim H (Y ) = p 1. Then there is a fibre sequence ΩĀmin (Σ b Y p+1 Y ) P Ē(Y ) E Ā min (Y ) H p Ā min (Σ b Y p+1 Y ). with the following properties: 1) On mod p homology H (Ē(Y )) = E(Σ 1 H (Y )) as coalgebras. 2) H (Āmin (Y )) = H (Ē(Y )) H (Āmin (Σ b Y p+1 Y )) as coalgebras. 3) If f : S n Y is a co-h map such that f 0: H (S n ) H (Y ). Then there is a commutative diagram of fibre sequences ΩĀmin (Σ b Y p+1 Y ) P Ē(Y ) ΩĀmin (Σ b Y p+1 P f Y ) S n 1 E Ā min (Y ) Hp Ā min (Σ b Y p+1 Y ) Bf Ā min (Σ b Y p+1 Y ). In particular, the map P : ΩĀmin (Σ b Y p+1 Y ) Ē(Y ) factors through the bottom cell of Ē(Y ).

5 REPRESENTATION THEORY IN HOMOTOPY AND EHP SEQUENCES 5 The following theorem gives a general criterion when the EHP fibration splits off. A space X is called to have a retractile generating complex C if C is a retract of ΣX with a retraction r : ΣX C such that the mod p cohomology H (X) is generated by M = Im(Σ 1 r : Σ 1 H (C) H (X)) and M = QH (X), the set of indecomposables. Recall that a space X is called (stably) atomic if any self (stable) map of X inducing isomorphism on the bottom homology is a (stable) homotopy equivalence. Theorem 1.6. Let p > 2 and let Y be any p-local simply connected co-h space such that Y is stably atomic. Suppose that H odd (Y ) = 0 and dim H (Y ) = p 1. Then the following statements are equivalent to each other: 1) The EHP fibration Ē(Y ) E Ā min (Y ) H p Ā min (Σ b Y p+1 Y ) splits off. 2) Ē(Y ) is an H-space. 3) There exists an H-space X having Y as a retractile generating complex. 4) The map P : ΩĀmin (Σ b Y p+1 Y ) Ē(Y ) is null homotopic. 5) The composite Σ b Y p 1 Y ΩĀmin (Σ b Y p+1 Y ) P Ē(Y ) is null homotopic. 6) There exists a map g : Σ b Y p Y Āmin (Y ) such that g : H (Σ b Y p Y ) H (Āmin (Y )) is a monomorphism on the bottom cells of Σ b Y p Y. According to [8], the space Ē(Y ) is not a retract of ΩY provided that p = 3 and Y = S 2k α1 e 2k+4 with 2k > 4 and 2k 1 2 mod 3. In this case one can see that Ā min (Y ) is atomic and so the EHP fibrations do not split off in general. It is an open classical problem whether a finite complex Y can be a retractile generating complex of a (finite) H-space X. The understanding of this problem helps to classify finite H-spaces. From the assumption that Y is a retract of ΣX, the space Y must be a co-h-space provided that Y is a retractile generating complex of an H-space. For p > 2, Hodd (Y ) must be zero provided that Y is a retractile generating complex of a finite H-space. Thus the co-h-spaces Y with H odd (Y ) = 0 are the interesting cases in this problem. It was proved in [6, 4, 20] that the problem has the positive answer for p-local simply connected co-h-space Y with the size dim H (Y ) p 2. The next interesting cases are of course for (p 1)-cell co-h-spaces. By thinking the spheres as (p 1)-cell complexes for p = 2, the classical Hopf invariant one problem was asking when an odd sphere is an H-space. When p = 3, it is an open classical problem when the total space of a spherical bundle over an odd sphere with fibre a odd sphere is an H-space [8, 10, 15]. Theorems 1.5 and 1.6 provide some new information for studying these open classical problems for p 3. We give some historic remarks on natural coalgebra decompositions of tensor algebras. Classically there have been a lot of study on decompositions of loop spaces for individual spaces. Particularly as a consequence of the study of decompositions of the loop spaces of Moore spaces, Cohen-Moore-Neisendorfer were able to

6 6 J. WU determine the best exponent of the homotopy groups of the spheres in odd prime cases [3]. It was discovered in [5] that the compositions of Hopf invariants with the loop of Whitehead products induce decompositions of loop suspensions with applications to mod 2 Moore spaces. It was then found that it is extremely difficult for having explicit constructions of finest retracts of the functor ΩΣ and so a systematic study on functorial decompositions of the functor ΩΣ came out naturally in [20] with an important discovery that the functorial decompositions of the functor ΩΣ are one-to-one correspondent to the natural coalgebra decompositions of T : V T (V ) as a functor from modules to coalgebras, where T (V ) is Hopf by saying V primitive. Thus the problem on functorial decompositions of loop suspensions (and more generally of loops on co-h-spaces) is reduced to the algebraic problem on natural coalgebra decompositions of tensor algebras. There have been various results in [20] including a solution to the Cohen conjecture. The new progress in this article is to give several machinery tools for obtaining natural coalgebra retracts of the functor T. Theorem 2.9 gives a general criterion how to check a sub Hopf algebra functor of T to be a natural coalgebra retract. Theorem 2.11 states that the sub Hopf algebra generated by a natural coalgebra retract of T is also a natural coalgebra retract of T. This is a very useful result. There have many natural coalgebra retracts of T with explicit information on primitive elements and so, by using this theorem, there are many explicit natural sub Hopf algebras of T that are natural coalgebra retracts of T. Theorem 2.12 gives many of these examples. By taking the geometric realization, one gets many explicit decompositions of the loops on co-h-spaces. Theorem 2.15 can be regarded as one of the fundamental theorems in this topic. From the definition, the functors A min and B max are only unique up to natural equivalence. Theorem 2.15 states that B max is the unique sub functor of T. Namely any natural coalgebra retract of T, which is naturally equivalent to B max, must be identically the same as B max as the sub functors of T. As a surprising consequence, Corollary 2.17 states that there are NO nontrivial natural coalgebra transformations from B max to A min. From this, Theorem 2.20 determines the irreducible factors of the associated Lie modules Lie min (n) over the symmetric group algebra k(σ n ) of the functor A min corresponding to two-row Young diagrams. As it is described in Theorem 2.18, the functor A min is determined by its associated Lie modules. Thus the determination of the irreducible factors of Lie min (n) helps to determine the functor A min. There seems a possibility for determining irreducible factors of Lie min (n) by constructing as many as possible factors in B max by the rule given in Corollary The geometry of natural coalgebra decompositions of the functor T is discussed in Section 4. Theorem 4.3 gives the geometric realization theorem for any natural coalgebra retract of T. The suspension splitting theorem for the geometric realizations of natural coalgebra retract of T is given in Theorem 4.8. Theorem 4.9 states that the geometric realization of any natural coalgebra-split sub Hopf algebras of T are loop spaces. The James-Hopf invariants can be defined for the loop spaces of any simply connected spaces with the property described in Theorem From this, the James-Hopf invariants for the geometric realization of any natural coalgebra retracts of T are obtained. From these progress, it seems that the notion of representation theory in homotopy comes out naturally. Also it gives a tight relation between the modular representation theory of the symmetric groups and the unstable homotopy theory.

7 REPRESENTATION THEORY IN HOMOTOPY AND EHP SEQUENCES 7 By taking the dual Hopf algebra T of the tensor algebra functor T, one obtains the shuffle algebras. Thus the natural coalgebra decompositions of T is equivalent to natural algebra decompositions of the shuffle algebra functor T. The functor A min is given as the smallest natural coalgebra retract of T over the polynomial algebra functor P. Thus the dual A min is the smallest natural algebra retract of T containing the functor Γ = P. In other words, one could start with the functor Γ and do the algebra extension of Γ in T by adding more generators until it reaches the smallest algebra retract A min of T. This gives some connections to the Galois theory. From the ideas arising from Galois theory, it is expected to have some connections between self natural algebra transformations of T over Γ and the functor A min. On the other hand, the group of self natural algebra transformations of T is related to the classical exponent problem in homotopy theory with connections to the complicity of algorithms [28]. Further exploration on the Galois theory on the functor T will be given in our subsequent articles. Since natural coalgebra retracts of the functor T are controlled by its associated Lie modules, there are canonical connections to the representation theory of free Lie modules over the general linear groups by taking the evaluations of the functors on a given module. By using Schur algebra [17], there are strong connections between the modular representation theory of the symmetric groups and the representation theory of self tensor products over the general linear groups. The connections between natural coalgebra retracts of the functor T and the free Lie modules over the general linear groups become less obvious. Namely there are some naturally indecomposable coalgebra retracts of T that become decomposable over the general linear groups by evaluating on some particular modules (for instance the modules given in Theorem 1.1). This gives a mystery between Lie type modules over the symmetric groups and free Lie modules over the general linear groups. It deserves to explore further on the connections between these Lie type modules. Finally it should be pointed out that the representation theory on natural coalgebra decompositions of the functor T may be applied to the Adams spectral sequences by using simplicial groups. Recall that the Adams spectral sequences can be obtained from the mod p descending central series of free simplicial groups. From Theorem 4.9, the geometric realization of any natural coalgebra-split sub Hopf algebras of T can be given as the simplicial sub group of Kan s G-construction on co-h-spaces Y. By taking the induced filtration from the mod p descending central series of GY, one obtains a filtered decomposition of GY. As a consequence, the Adams spectral sequences are decomposed into different pieces. The decompositions of the Adams spectral sequences help to handle the differentials in the spectral sequences. Moreover there seems a possibility for having applications to the Goodwillie towers by establishing simplicial group models for Goodwillie towers. It seems expected that the Goodwillie towers can be decomposed into many pieces in which one atomic piece corresponds to the functor A min having some properties similar to the Goodwillie towers for the spheres such as the n-adic terms in the Goodwillie towers of the functor A min might be removable if n is not a power of p. The exploration on these further possible applications will be given in our subsequent articles. The article is organized as follows. In section 2, we study the representation theory on natural coalgebra decompositions of tensor algebras. The proof of Theorem 1.1 is given in Section 3. The geometry of natural coalgebra decompositions

8 8 J. WU of tensor algebras is investigated in Section 4, where Theorem 1.4 is Theorem 4.8. In Section 5, we give the proof of Theorem 1.5. The proof of Theorem 1.6 is given in Section Representation Theory of The Tensor Algebra Functor 2.1. The Functor A min on Ungraded Modules. In this subsection, the ground ring is a field k. A coalgebra means a pointed coassociative cocommutative coalgebra. For any module V, the tensor algebra T (V ) is a Hopf algebra by requiring V to be primitive. This defines T : V T (V ) as a functor from ungraded modules to coalgebras. The functor A min : V A min (V ) is defined to be the smallest coalgebra retract of the functor T with the property that V A min (V ). More precisely the functor A min is defined by the following property: (1). A min is a functor from k-modules to coalgebras with a natural linear inclusion V A min (V ). (2). A min (V ) is a natural coalgebra retract of T (V ) over V. Namely there exist natural coalgebra transformations s: A min T and r : T A min such that the diagram A min (V ) sv T (V ) r V A min (V ) V ========= V ========= V commutes for any V and r s = id A min. (3). A min is minimal with respect to the above two conditions: if A is any functor from k-modules to coalgebras with natural linear inclusion V A(V ) such that A(V ) is a natural coalgebra retract of T (V ) over V, then A min (V ) is a natural coalgebra retract of A(V ) over V. By the minimal assumption, the functor A min is unique up to natural equivalence if it exists. The existence of the functor A min follows from [20, Theorem 4.12]. There is a multiplication on A min given by the composite A min A min s s T T µ T r A min and so A min is a functor from modules to quasi-hopf algebras, where a quasi-hopf algebra means a coassociative and cocommutative bi-algebra without assuming the associativity of the multiplication. The multiplication on A min induces a new natural coalgebra transformation r V : T (V ) Amin (V ) given by r V (x 1 x n ) = ((( (x 1 x 2 ) x 3 ) ) x n ) for any x 1 x n V n. By the minimal assumption, the composite A min s T r A min is a natural equivalence. Consider T (V ) as an A min (V )-comodule via the map r V. According to [20, Proposition 6.1], the cotensor product B max (V ) = k A min (V )T (V )

9 REPRESENTATION THEORY IN HOMOTOPY AND EHP SEQUENCES 9 is natural sub Hopf algebra of T (V ) with a natural coalgebra equivalence (2.1) k B max (V ) T (V ) = A min (V ). Together with [20, Lemmas 5.3], there is a natural coalgebra decomposition (2.2) T (V ) = B max (V ) A min (V ) The Symmetric Group Module Lie(n). Let the ground ring R be any commutative ring with identity. The module Lie R (n), which is simply denoted as Lie(n) if the ground ring is clear, is defined as follows. Let V be a free R-module of rank n with a basis {e i 1 i n}. Define the module γ n is defined to be the submodule of V n spanned by e σ(1) e σ(2) e σ(n) for σ Σ n. The module Lie(n) is defined by Lie(n) = γ n L n ( V ) V n. The Σ n -action on γ n is given by permuting letters e 1, e 2,..., e n. Since both γ n and L n ( V ) are invariant under permutations of the letters, Lie(n) is an R(Σ n )- submodule of γ n. Note that Lie(n) is the submodule of L n ( V ) spanned by the homogenous Lie elements in which each e i occurs exactly once. By the Witt formula, Lie(n) is a free R-module of rank (n 1)!. Following from the antisymmetry and the Jacobi identity, Lie(n) has a basis given by the elements [[e 1, e σ(2) ], e σ(3) ],..., e σ(n) ] for σ Σ n 1. (See [2].) Observe that Lie(n) is the image of the R(Σ n )-map β n : γ n γ n β n (a 1 a n ) = [[a 1, a 2 ],..., a n ]. Thus Lie(n) can be also regarded as the quotient R(Σ n )-module of γ n with the projection β n : γ n Lie(n). Proposition 2.1. Let V be any graded projective module and let Σ n act on V n by permuting factors in graded sense. Then there is a functorial isomorphism for any graded module V. Lie(n) R(Σn) V n = Ln (V ) Proof. Clearly the quotient map β n : V n = γ n R(Σn) V n L n (V ), x 1 x n [[x 1, x 2 ],..., x n ], factors through the quotient Lie(n) R(Σn) V n and so there is an epimorphism Lie(n) R(Σn) V n L n (V ). On the other hand note that L = Lie(n) R(Σn) V n has the canonical graded Lie algebra structure generated by V. So the inclusion V L induces an epimorphism of graded Lie algebras L(V ) L. The assertion follows. Consider T n : V V n as a functor from projective (ungraded) modules to projective (ungraded) modules. Denote by Hom(F, F ) the set of natural linear transformations from a functor F to a functor F provided that F preserves direct limits. By [9, Lemma 3.8], Hom(T n, T m ) = 0 if n m and there is an isomorphism of rings (2.3) θ : End R(Σn)(γ n ) End(Tn, T n )

10 10 J. WU given by θ(φ) = φ idv n : γ n R(Σn) V n = V n γ n R(Σn) V n = V n for φ End R(Σn)(γ n ). Replacing γ n by Lie(n), we have the morphism of rings θ : End R(Σn)(Lie(n)) End(Ln ). Proposition 2.2. If n m, then Hom(L n, L m ) = 0. Moreover the map is an isomorphism. θ : End R(Σn)(Lie(n)) End(L n ) Proof. Let φ: L n L m be a natural transformation. Then the composite T n Ln φ Lm Tm is a natural transformation, which is zero as Hom(T n, T m ) = 0. Thus φ = 0. For the second statement, clearly θ is a monomorphism. Let φ V : L n (V ) L n (V ) be any natural transformation. Let V be the free R-module of rank n, which defines γ n. Consider the commutative diagram T n ( V ) β n Ln ( V ) φ V Ln ( V ) T n ( V ) γ n β n Lie(n) φ Lie(n) γn, where the existence of φ follows from the fact that the composite of the maps in the top row maps γ n into γ n and Lie(n) = γ n L n ( V ). Let V be any ungraded module and let a 1 a n be any homogenous element in V n. Let f : V V be R-linear map such that f(e i ) = a i. By the naturality, there is a commutative diagram Lie(n) L n ( V ) L n(f) Ln (V ) φ φ V φ V Lie(n) Ln ( V Ln (f) ) Ln (V ) Thus θ(φ )([[a 1, a 2 ],..., a n ]) = φ([[a 1, a 2 ],..., a n ]) and hence the result. Corollary 2.3. There is a one-to-one correspondence between the decompositions of the functor L n and the decompositions of Lie(n) over R(Σ n ). A functor from modules to modules Q is called T n -projective if Q is naturally equivalent to a retract of the functor T n. Proposition 2.4. Let Q be a T n -projective functor and let φ: Q L n be a natural linear transformation. Then there exists a natural linear transformation φ: Q T n such that φ = β n φ.

11 REPRESENTATION THEORY IN HOMOTOPY AND EHP SEQUENCES 11 Proof. Let V be the free R-module of rank n, which defines γ n. Let Q = γ n Q( V ). Since Q is a retract of the functor T n, Q is a summand of γ n over R(Σ n ) and so Q is a projective R(Σ n )-module. From the fact that End R(Σn)(γ n ) = End(T n ), Q(V ) = Q R(Σn) V n for any module V. By the proof of Proposition 2.2, the map θ : Hom R(Σn)( Q, Lie(n)) Hom(Q, L n ), f f id V n is an isomorphism. Thus there exists a unique R(Σ n )-map φ : Q Lie(n) such that θ(φ ) = φ. Since Q is projective, the lifting problem γ n φ β n φ Q Lie(n) has a solution. The assertion follows by tensoring with V n over R(Σ n ). By inspecting the proof, each T n -projective sub functor Q of L n induces a R(Σ n )- projective submodule Q of Lie(n). Conversely, each R(Σ n )-projective submodule Q of Lie(n) induces a T n -projective sub functor Q, V Q R(Σn) V n, of L n. Thus we have the following: Proposition 2.5. There is a one-to-one correspondence between T n -projective sub functors of L n and R(Σ n )-projective submodules of Lie(n) Representation Theory of Tensor Algebra Functor. Some results in the representation theory help us to change the ground ring. Let Z (p) be the p-local integers. By modular representation theory of symmetric groups (see, for example [7, Exercise 6.16, p.142]), any idempotent in (Z/p)(Σ n ) lifts to an idempotent in Z (p) (Σ n ). It is well-known [11] that any irreducible modules M over Z/p(Σ n ) is absolutely irreducible, that is, for any extension field k, M k is irreducible over k(σ n ). Thus there is a one-to-one correspondence between idempotents in Z/p(Σ n ) and the idempotents in k(σ n ). Let R be any commutative ring with identity. Consider T : V T (V ) as the functor from projective R-modules to coalgebras over R. Denote by coalg R (T, T ) the set of self natural coalgebra transformation of T. Let k be any field of characteristic p. We have the canonical functions by modulo p and R: coalg Z (p) (T, T ) coalg Z/p (T, T ) K : coalg Z/(p) (T, T ) coalg k (T, T ) by tensoring with k over Z/p. By [20, Corollary 6.9], there is a one-to-one correspondence between natural indecomposable retract of T over k and the indecomposable k(σ n )-projective submodule of Lie k (n) for n 1. Thus we have the following: Proposition 2.6. The functions R and K have the following properties:

12 12 J. WU (1). The map R: coalg Z (p) (T, T ) coalg Z/p (T, T ) induces a one-to-one correspondence betweens idempotents. Thus every natural coalgebra decompositions of T over Z/p lifts to a natural coalgebra decomposition over Z (p). (2). The map K : coalg Z (p) (T, T ) coalg Z/p (T, T ) induces a one-to-one correspondence betweens idempotents. Thus natural coalgebra decompositions of T only depends on the characteristic of the ground field. By this proposition, we can freely change the ground fields k with the same characteristic and lift natural coalgebra decompositions over p-local integers if it is necessary. Let C be a sub functor of T. Let C n = C T n : V C(V ) T n (V ) be the homogenous component of C of tensor length n. Lemma 2.7. Let C(V ) be a natural sub coalgebra of T (V ) such that C is a natural coalgebra retract of T. If C j = 0 for 0 < j < n, then C n (V ) L n (V ) for every V. Proof. Since C is a natural coalgebra retract of T, we may assume that the ground ring is Z (p). From the assumption that C j = 0 for 0 < j < n, C n (V ) P T (V ) = L(V ), where P T (V ) is the set of the primitives of T (V ), and so C n (V ) L n (V ). Lemma 2.8. Let Q be a T n -projective sub functor of L n. Then the sub Hopf algebra T (Q(V )) of T (V ) generated by Q(V ) is a natural coalgebra retract of T (V ). Proof. Since Q is a T n -projective sub functor of L n, there exists a (unique) k(σ n )- projective submodule Q of Lie(n) such that Q(V ) = Q k(σn) V n. Since Q is a finite dimensional projective over the group algebra k(σ n ), it is an injective module over k(σ n ) and so the inclusion Q Lie(n) γ n admits a retraction r : γ n Q over k(σ n ). By tensoring with V n over k(σ n ), we have the natural linear transformation r v = r id V n : V n = γ n k(σn) V n with r V Q(V ) = id Q(V ). Let H n : T (V ) T (V n ) Q(V ) = Q k(σn) V n be the algebraic James-Hopf map induced by the geometric James-Hopf map by taking homology. Then there is a commutative diagram T (Q(V )) T (L n (V )) T (V ) ============= T (j V ) T (Q(V )) T (r V ) H n T (V n ), where the maps in the top row are the inclusions of sub Hopf algebras, j V is the canonical inclusion and the right triangle commutes by the geometric theorem in [26, Theorem 1.1]. Thus the sub Hopf algebra T (Q(V )) of T (V ) admits a natural coalgebra retraction and hence the result.

13 REPRESENTATION THEORY IN HOMOTOPY AND EHP SEQUENCES 13 A natural sub Hopf algebra B(V ) of T (V ) is called coalgebra-split if the inclusion B(V ) T (V ) admits a natural coalgebra retraction. For a Hopf algebra A, denote by QA the set of indecomposable elements of A. Let IA be the augmentation ideal of A. If B(V ) is a natural sub Hopf algebra of T (V ), then there is a natural epimorphism IB(V ) QB(V ). Let Q n B(V ) be the quotient of B n (V ) = IB(V ) T n (V ) in QB(V ). Theorem 2.9. Let B(V ) be a natural sub Hopf algebra of T (V ). Then the following statements are equivalent to each other: (1). B(V ) is a natural coalgebra-split sub Hopf algebra of T (V ). (2). There is a natural linear transformation r : T (V ) B(V ) such that r B(V ) is the identity. (3). Each Q n B is naturally equivalent to a T n -projective sub functor of L n. (4). Each Q n B is a T n -projective functor. (5). There exists a finite or countable collection of T li -projective functors Q li such that B(V ) is generated by { Q li (V )} i I as an algebra. Note. In (5), Q li is NOT required to be a sub functor of the Lie functor L li. Also we do not require that the collection { Q li } is linearly independent or algebraically independent. But we require that the subalgebra B(V ) must be a sub Hopf algebra of T (V ). Thus, for obtaining possible larger natural coalgebra-split sub Hopf algebra of T (V ), one can freely add more T -projective functors as new generators provided that the resulting new subalgebra is Hopf. Proof. (1) = (2), (3) = (4) and (4) = (5) are obvious. By [20, Theorem 8.6], (2) = (1). Thus (1) (2). From the proof of [20, Theorem 8.8], (2) = (3). (4) = (2). Since B(V ) is a sub Hopf algebra of primitively generated Hopf algebra T (V ), B(V ) is primitively generated and so r n : P n B(V ) = B(V ) L res n (V ) Qn B(V ) is a natural epimorphism, where L res (V ) = P T (V ) is the free restricted Lie algebra generated by V. We first prove that the map r n admits a natural cross-section s n : Q n B(V ) P n B(V ). Let Q n be the k(σ n )-projective module such that Q n B(V ) = Q n k(σn) V n. Since r n : P n B(V ) Q n B(V ) is a natural epimorphism, it induces a simplicial epimorphism r n : P n B(V ) Q n B(V ) and so an epimorphism of Moore chain complex N(r n ): NP n B(V ) NQ n B(V ) for any simplicial module V. Let V = { V n } n 0 = K(k, 1) be the simplicial k-module given by: 1) dim V n = n with the basis {e 1,..., e n }. 2) the faces d i : V n V n 1, 0 i n, is defined by the formula: e j if 1 j < i, 0 if j = i, d i e j = e j 1 if 0 i < j, j > 1, (e 1 + e e n 1 ) if i = 0, j = 1. 3) the degeneracy s i : V n V n+1, 0 i n, is defined by the formula: e j if 1 j < i, s i e j = e i + e i+1 if j = i, e j+1 if 0 i < j.

14 14 J. WU Then it is straightforward to check that NP n B( V n ) = Ker(d i ) = B( V ) L res ( V ) γ n = B( V ) Lie(n), i=1 NQ n B( V ) = Q n. Since Q n is k(σ n )-projective, the k(σ n )-epimorphism N(r n ): NP n B( V ) = B( V ) Lie(n) Qn admits a k(σ n )-cross-section. By tensoring with V n over k(σ n ), we have a natural cross-section s n : Q n B(V ) = Q n k(σn) V n (B( V ) Lie(n)) k(σn) V n B(V ) Ln (V ) B(V ) L res n (V ) to the natural quotient map r n : B(V ) L res n (V ) Q n B(V ). Now we show that the inclusion B(V ) T (V ) admits a natural linear retraction. By identifying Q n B(V ) with s n (Q n B(V )), we have ( ) B(V ) = T Q k B(V ) T (V ). k=1 Since each Q k B is a retract of the functor T k, Q i1 B Q it B is a retract of T i1+i 2+ +i t for any sequence (i 1,..., i t ). Consider Q i1 B(V ) Q i2 B(V ) Q it B(V ) V q. i 1+i 2+ +i t=q Let V be a q-dimensional module which defines γ q. Since {Q i B( V ) i 1} are algebraically independent, the summation ( Qi1 B( V ) Q i2 B( V ) Q it B( V ) ) γ q γ q i 1+i 2+ +i t=q is the direct sum of the k(σ q )-projective modules ( Qi1 B( V ) Q i2 B( V ) Q it B( V ) ) γ q. Thus there is a k(σ q )-retraction ( φ: γ q Qi1 B( V ) Q i2 B( V ) Q it B( V ) ) γ q. i 1+i 2+ +i t=q By tensoring with V q over k(σ q ), there is natural linear retraction V q Q i1 B(V ) Q i2 B(V ) Q it B(V ) i 1+i 2+ +i t=q for any q 1. Thus the inclusion B(V ) T (V ) admits a natural linear retraction. (5) = (1). We may assume that Define l 1 l 2 l 3 B [m] (V ) = Q li (V ) 1 i m be the sub algebra of T (V ) generated by Q li (V ) with 1 i m.

15 REPRESENTATION THEORY IN HOMOTOPY AND EHP SEQUENCES 15 First we show by induction that each B [m] (V ) is a natural coalgebra-split sub Hopf algebra. When m = 1, since Q l1 (V ) has the bottom tensor length in B(V ), Q l1 (V ) P B l1 and B [1] (V ) is a sub Hopf algebra of T (V ). By lemma 2.8, B [1] is coalgebra split. Assume that the statement holds for m 1. Since the inclusion B [m 1] (V ) T (V ) factors through B(V ), B [m 1] (V ) is a natural coalgebra retract of B(V ) and so there is a natural decomposition natural decomposition B(V ) = B [m 1] (V ) (k B [m 1] (V ) B [m] (V )). Let r : B(V ) B [m 1] (V ) be a natural coalgebra retraction. Then Ker(r) P B(V ) lm because B [m 1] (V ) j B(V ) j is an isomorphism for j < l m. Let D lm (V ) be the kernel of the composite Q lm (V ) B(V ) r B [m 1] (V ). Then D lm (V ) P B(V ) lm Thus B [m] (V ) is the sub algebra of T (V ) by D lm (V ) and B [m 1] (V ), and so B [m] (V ) is a sub Hopf algebra of T (V ). Note that the composite D lm (V ) Qlm (V ) B [m] (V ) lm ( k B [m 1] (V ) B [m] (V ) ) l m = Q lm B [m] (V ) is an isomorphism. Thus Q lm B [m] is T lm -projective functor as it is a retract of Q lm. By induction, B [m 1] (V ) is coalgebra-split. Since we already proved (1) (4), each Q i B [m 1] is T i -projective. Note that QB [m] = Q lm B [m] QB [m 1]. Each Q i B [m] is also T i -projective and so B [m] is coalgebra-split by (4). The induction is finished. Now for each n, there exists m n such that Q n B = Q n B [mn] because B is the union of the increasing sequence of the sub Hopf algebras B [m]. Thus Q n B is T n - projective for each n and so B is coalgebra-split by (4). The proof is finished. The following Corollary gives an algorithm that if we know a natural coalgebrasplit sub Hopf algebra B (V ) of T (V ) and a T n -projective sub functor Q n of L n, then we obtain a larger natural coalgebra-split sub Hopf algebra of T (V ) by adding Q n. Corollary Let B (V ) be a natural coalgebra-split sub Hopf algebra of T (V ) and let Q n be a T n -projective sub functor of L n. Then the subalgebra B(V ) = B (V ), Q n (V ) of T (V ) generated by B (V ) and Q n (V ) is a natural coalgebra-split sub Hopf algebra of T (V ). Note. Here the assumption the Q n is sub functor of L n is used for making sure that B(V ) is a sub Hopf algebra of T (V ).

16 16 J. WU Proof. Since B (V ) is a natural sub Hopf algebra of T (V ) and Q n (V ) L n (V ), B(V ) is a natural sub Hopf algebra of T (V ). By Theorem 2.9, each Q i B is T i - projective. Observe that The assertion follows from Theorem 2.9. B(V ) = Q i B (V ), Q n (V ) i 1. The following theorem gives an important property that every natural coalgebra retract of T generates a natural coalgebra-split sub Hopf algebra of T. Theorem Let C(V ) be a natural sub coalgebra of T (V ) such that C is a natural coalgebra retract of T. Let B(V ) = C(V ) be the subalgebra of T (V ) generated by C(V ). Then B(V ) is a natural coalgebra-split sub Hopf algebra of T (V ). Moreover B(V ) = C(V ) L(V ). Note. Since C is a retract of T, each C n is T n -projective. For applying Theorem 2.9, we need to check that B(V ) is a sub Hopf algebra of T (V ). Proof. The second assertion follows from the first assertion because by Theorem 2.9 any natural coalgebra-split sub Hopf algebra of T (V ) are generated by Lie elements. Let B [n] (V ) = C i (V ) 1 i n be the subalgebra of T (V ). Since C(V ) is a natural retract of T (V ), each C n is T n -projective. By Theorem 2.9, it suffices to show by induction that B [n] (V ) is a sub Hopf algebra of T (V ). Since C 1 (V ) L 1 (V ), B [1] (V ) is a sub Hopf algebra of T (V ). Suppose that B [n 1] (V ) is sub Hopf algebra of T (V ). By Theorem 2.9, B [n 1] (V ) is a natural coalgebra-split sub Hopf algebra of T (V ). Let C(V ) be the colimit of the coalgebra map T (V ) retraction B [n 1] (V ) T (V ) retraction C(V ) T (V ). By [20, Theorem 4.5], C(V ) is a common coalgebra retract of T (V ), B [n 1] (V ) and C(V ) with C j (V ) = C j (V ) for j < n because C j (V ) B [n 1] j (V ). Since C is a natural coalgebra retract of T, there is a natural multiplication on C(V ) given by the composite C(V ) C(V ) T (V ) T (V ) µ T (V ) C(V ) and so C is a functor from modules to quasi-hopf algebras. By [20, Lemma 5.2], there exists a natural coalgebra retract C of C with natural coalgebra decomposition C(V ) = C(V ) C (V ). Since C j (V ) = C j (V ) for j < n, C j (V ) = 0 for 0 < j < n and By Lemma 2.7, C n(v ) L n (V ). Since C n (V ) = C n (V ) C n(v ). C(V ) B [n 1] (V ), the map C n(v ) = C n (V )/ C n (V ) Cn (V )/(C n (V ) B [n 1] n (V ))

17 REPRESENTATION THEORY IN HOMOTOPY AND EHP SEQUENCES 17 is onto and so B [n] (V ) is generated by B [n 1] (V ) and C n(v ), which is a sub Hopf algebra of T (V ) because C n(v ) L n (V ). The induction is finished and hence the result. By using this theorem, we are able to construct many explicit natural coalgebrasplit sub Hopf algebras of T (V ) and so one will get many explicit functorial decompositions of loops on co-h-spaces after taking geometric realizations. Theorem Let {m i } be the set of positive integers prime to p with 1 < m 1 < m 2 <. Then the sub Hopf algebra B(V ) of T (V ) generated by L mipr(v ) for i 1, r 0 is natural coalgebra-split. In particular, the sub Hopf algebra of T (V ) generated by is natural coalgebra split. L n (V ) for n NOT a power of p Proof. According to [19, Theorem 4.6], there is a natural coalgebra retract C(V ) such that the primitives { P Tn (V ) if n = m P C n (V ) = i p r for some i, r 0 0 otherwise. Let B(V ) = C(V ). By Theorem 2.11, B(V ) = C(V ) L(V ) is a natural coalgebra-split sub Hopf algebra of T (V ) and hence the result. Example For p = 2 and m = 3, we have natural coalgebra-split sub Hopf algebra L 3 (V ), L 6 (V ), L 12 (V ), L 24 (V ),... of T (V ). By Proof of Theorem 2.9, L 3 (V ), L 6 (V ),..., L 3 2 r(v ) is also a natural coalgebra-split sub Hopf algebra of T (V ) for each r 0. determining indecomposables, L 6 = L 6 /L 2 (L 3 ) is a T 6 -projective sub functor of L 6. Similarly L 12 = L 12 / ([L 6, L 6] [[L 6, L 3 ], L 3 ] L 4 (L 3 )) is a T 1 2-projective sub functor of L 12. In general, L 3 2 r = L 3 2 r/ (L(L 3, L 6, L 1 2,..., L 3 2 r 1) L 3 2 r) is a T 3 2 r-projective sub functor of L 3 2 r. Thus one obtains a sequence k(σ 3 2 r)- projective submodules of Lie(3 2 r ). Let Lie max (n) denote the maximal k(σ n )-projective submodule of Lie(n). More precisely the module Lie max (n) is defined by the following property: (1). Lie max (n) is a k(σ n )-projective submodule of Lie(n); (2). Lie max (n) is maximal with respect to property (1). Namely any k(σ n )- projective submodule of Lie(n) is isomorphic to a k(σ n )-summand of Lie max (n). By

18 18 J. WU According to [20, Theorem 7.4], the module Lie max (n) exists with dimlie max (n) = max{dimp P is a k projective submodule of Lie(n)} is independent on the choice of Lie max (n). Proposition As a submodule of Lie(n), Lie max (n) = {P Lie(n) P is k(σ n ) projective}. Thus Lie max (n) is the unique submodule of Lie(n). Proof. Let Lie max (n) be a fixed choice of maximal k(σ n )-projective submodule of Lie(n). Let P be any k(σ n )-projective submodule of Lie(n) and let M = Lie max (n) + P Lie(n). Since Lie max (n) is a finite dimensional projective module over the group algebra k(σ n ), Lie max (n) is injective over k(σ n ) and so the inclusion Lie max (n) M = Lie max (n) + P admits a k(σ n )-retraction r. Let Q be the kernel of the composite Then the composite P M r Lie max (n). Q P P/(P Lie max (n)) = (P + Lie max (n))/lie max (n) is an isomorphism. Thus Q is a summand of P and so Q is k(σ n )-projective. It follows that M = Lie max (n) Q is k(σ n )-projective. From the inequality we have Lie max (n) = M. Thus dimlie max (n) dimm dimlie max (n), P Lie max (n) = M and hence the result. Thus Lie max (n) is given by adding all possible k(σ n )-submodules of Lie(n). Let V be any module. Define (2.4) L max n (V ) = Lie max (n) k(σn) V n. Thus we have the sub functor L max n of L n. By Proposition 2.14, we have (2.5) L max (V ) = {Q(V ) L n (V ) Q is T n projective}. Theorem 2.15 (Uniqueness Theorem of B max ). Let B max (V ) be the sub Hopf algebra of T (V ) generated by L max n (V ) = Lie max (n) k(σn) V n for n 2. Then the following hold: (1). There is a natural coalgebra equivalence for any ungraded module V. k B max (V ) T (V ) = A min (V )

19 REPRESENTATION THEORY IN HOMOTOPY AND EHP SEQUENCES 19 (2). B max is the unique largest coalgebra retract of T (V ) that does not V in the following sense: B max (V ) = {B(V ) T (V ) B is a natural coalgebra retract of T with B 1 = 0}. Proof. Assertion (1) follows from [20, (6) of Proposition 10.3]. (2). Clearly B max (V ) {B(V ) T (V ) B is a natural coalgebra retract of T with B 1 = 0}. Let B(V ) T (V ) be any natural coalgebra retract of T (V ) such that B 1 = 0. By Theorem 2.11, B(V ) = B(V ) is natural coalgebra-split sub Hopf algebra of T (V ) with B 1 = 0 because B 1 = 0. By Theorem 2.9, Q n B(V ) is naturally equivalent to a Tn -projective sub functor of L n for each n. We can identify Q n B with its image in Bn L n. Since B 1 = 0, we have Q B 1 = 0. From Equation (2.5), Q n B(V ) L max n (V ) for each n 2 and so B(V ) is a sub Hopf algebra of B max (V ). It follows that and hence the result. B(V ) B(V ) B max (V ) Corollary Let f : T (V ) A min (V ) be natural coalgebra transformation such that f V : T 1 (V ) = V A min 1 (V ) = V is a natural isomorphism. Then f B max : B max (V ) A min (V ) is the trivial natural coalgebra transformation for any V. Proof. Since A min is a natural coalgebra retract of T, there is a natural coalgebra injection s: A min T. By the minimal assumption of A min, the composite A min s T f A min is a natural equivalence. Let B(V ) be the cotensor product B(V ) = k A min (V )T (V ) by considering T (V ) as an A min (V )-comodule via the map f. Note that f B : B(V ) A min (V ) is the trivial map for any V. It suffices to show that B(V ) = B max (V ). By [20, Lemma 5.3], there is a natural coalgebra decomposition T (V ) = A min (V ) B(V ). In particular, B is a natural coalgebra retract of T with B 1 = 0. By Theorem 2.15, B(V ) B max (V ) for any V. For any finite dimensional module V, the Poincaré series χ( B(V )) = χ(t (V )) χ(a min (V )) = χ(bmax (V )).

20 20 J. WU Thus B(V ) = B max (V ) for any finite dimensional module V. From the naturality, B(V ) = B max (V ) for any module V and hence the result. Corollary Any natural coalgebra transformation φ: B max (V ) A min (V ) is trivial. Proof. The assertion follows from the commutative diagram T B max = B max A min φ id A min A min A min mult A min ===== B max k φ A min k, =============== where the top row is a natural coalgebra transformation which induces an isomorphism T 1 = A min 1. Let V be the n-dimensional module which defines γ n. Let γ max n = Bn max ( V ) γ n be the k(σ n )-submodule of γ n. Since B max is a natural coalgebra retract of T, each Bn max is natural linear retract of T n. Thus γn max is a k(σ n )-projective submodule of γ n with the property that Bn max (V ) = γn max k(σn) V n for any V. Consider A min as a natural sub coalgebra of T with a natural coalgebra retraction T A min. Define Lie min (n) = A min n ( V ) Lie(n). From the natural coalgebra decomposition T (V ) = B max (V ) A min (V ) over p-local integers, there is a natural decomposition of primitives L(V ) = P B max (V ) P A min (V ) with P B max (V ) = B max (V ) L(V ) and P A min (V ) = A min (V ) L(V ). Thus there is natural decomposition L n (V ) = (B max (V ) L n (V )) (A min (V ) L n (V )) over any field k of characteristic p. It follows that there is a decomposition over k(σ n ) and so Lie(n) = (γ max n Lie(n)) Lie min (n) A min (V ) L n (V ) = Lie min (n) k(σn) V n for any module V. Write L min n (V ) for A min (V ) L n (V ). Theorem With the notations defined as above, we have the following:

21 REPRESENTATION THEORY IN HOMOTOPY AND EHP SEQUENCES 21 (1). There is a non-functorial coalgebra isomorphism A min (V ) = Λ( L min n (V )) for any module V, where Λ(W ) is the free commutative algebra generated by W. (2). Hom k(σn)(γn max, Lie min (n)) = 0 for each n 1. Proof. Since the natural coalgebra decomposition T (V ) = B max (V ) A min (V ) can be lifting to over p-local integers, assertion (1) follows from the classical Poincaré- Birkhoff-Witt Theorem. (2). Let φ: γn max Lie min (n) be a k(σ n )-map. Define φ = φ id V n : Bn max (V ) = γn max k(σn)v n Lie min (n) k(σn)v n = L min n (V ). From [20, Proof of Theorem 8.6], there is a natural coalgebra transformation H n : B max (V ) T (Bn max (V )) such that the diagram B max (V ) Hn T (Bn max (V )) B max n (V ) commutes. Let f be the composite ==== Bn max (V ) B max (V ) Hn T (B max n (V )) T ( φ) T (L min n (V )) T (L n (V )) T (V ) rmin A min (V ), where r min is a natural coalgebra retraction. By Corollary 2.17, f is the trivial map for any V. In particular, f B max : B max n n (V ) A min n (V ) is zero for any V. Since r min is a natural coalgebra retraction, r min L min n (V ) : L min n (V ) A min n (V ) is the inclusion map. Thus f B max is the composite n B max n (V ) φ L min n (V ) A min n (V ) and so φ: Bn max (V ) L min n (V ) is zero for any V. Choose V = V. From the commutative diagram Bn max ( V φ ) L min n ( 0 V ) γ max n = Bn max ( V ) φ Lie min (n) = L min n ( V ) γ n,

22 22 J. WU we have φ = 0 and hence the result. Recall that up to conjugation primitive idempotents in Z (p) (S n ) are in one-to-one correspondence with p-regular partitions of n. A (proper) partition λ of n, denoted by λ n, is a sequence of positive integers λ = (λ 1,, λ s ) such that s λ 1 λ 2 λ s and λ j = n, where s is called the length of λ. A partition λ = (λ 1,, λ s ) is called p-regular if there is no subscript i with 1 i s such that λ i = λ i+1 = = λ i+p 1. For each p-regular partition λ, there is a Specht module S λ corresponding to λ. Let Q λ be a projective cover of S λ and let E λ = Hd(S λ ) be the head of S λ. Then {E λ } ({Q λ } ) give all irreducible (indecomposable projective) k(σ n )-modules when λ runs over all p-regular partitions of n.(see [7, 11] for details). From the above theorem that Hom k(σn)(γ max n Corollary Let Then Λ max n Λ max n Λ min n j=1, Lie min (n)) = 0, we have the following: = {λ n λ p-regular and Q λ is a summand of γ max n }, = {λ n λ p-regular and E λ is a factor of Lie min (n)}. min Λ n =. Note. It is possible that some partitions do not appear in both Λ max n and For instance, let p = 2, from [27, p. 44], Q (4,2) Λ max min 6 and Λ 6 =. Theorem The k(σ n )-module Lie min (n) has the following properties: (1). Lie min (n) = 0 if n is not a power of p. (2). E (ps) is an irreducible factor of Lie min (p s ). (3). E (ps 1,1) is an irreducible factor of Lie min (p s ). (4). E (ps k,k) is NOT an irreducible factor of Lie min (p s ) for k 2. Λ min n. Proof. Assertion (1) follows from Theorem 2.12 that L n (V ) B max (V ) if n is not a power of p. (2). Let V be the p s -dimensional module which defines γ p s. Let α = e σ(1) e σ(2) e σ(p s ). σ Σ p s Note that α generates the trivial k(σ p s)-module and α = [[e 1, e τ(2) ], e τ(3),..., e τ(p2 )] Lie(p s ). τ Σ p s 1 By [20, Corollary 8.18], for any natural coalgebra transformation f : T (V ) T (V ) such that f V = id V, f Tn( V )(α) = α. Thus α Lie min (p s ) = A min ( V ) Lie(p s ) and E (ps) is a factor of Lie min (p s ). (3). Let V be the n-dimensional module which defines γ n. Define Lie(n) = Lie(n)/ Lie(n) [L i ( V ), L j ( V )]. i + j = n i, j > 1